Wave packets, signalling and resonances in a homogeneous waveguide

We solve the initial-boundary-value linear stability problem for small localised disturbances in a homogeneous elastic waveguide formally by applying a combined Laplace-Fourier transform. An asymptotic evaluation of the solution, expressed as an inverse Laplace-Fourier integral, is carried out by means of the mathematical formalism of absolute and convective instabilities. Wave packets, triggered by localised in space and finite in time perturbations, as well as spatial responses to localised in space sources, with the time dependence satisfying e"-"i"#omega#"_0"t+O(e"-"#epsilon#"t), for t#->##infinity#, where Im #omega#_0=0 and #epsilon#>0, that is, the signalling problem, are treated. For this purpose, we analyse the dispersion relation of the problem analytically, and by solving numerically the eigenvalue stability problem. It is shown that due to double roots in a wavenumber k of the dispersion relation function D(k, #omega#), for real frequencies #omega#, that satisfy a collision criterion, wave packets with an algebraic temporal decay and signalling with an algebraic temporal growth, that is, temporal resonances, are present in a neutrally stable homogeneous waveguide. Moreover, for any admissible combination of the physical parameters, a homogeneous waveguide possesses a countable set of temporally resonant frequencies. Consequences of these results for modelling in seismology are discussed. (orig.)SIGLEAvailable from TIB Hannover: RR 6943(96/01) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman